{"paper":{"title":"Rational normal forms and stability of small solutions to nonlinear Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Benoit Grebert (LMJL), Erwan Faou (IRMAR, Inria, IRMAR), Joackim Bernier (MINGUS, MINGUS)","submitted_at":"2018-12-29T18:07:14Z","abstract_excerpt":"We consider general classes of nonlinear Schr\\\"odinger equations on the circle with nontrivial cubic part and without external parameters.  We construct  a new type of normal forms, namely rational normal forms, on open sets  surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\\varepsilon$, for generic initial data of size $\\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\\varepsilon^{M+1}$. This implies  that for such initial data $u(0)$ we con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.11414","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}